The Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of action in quantum mechanics. Published in 1900, it originally described the proportionality constant between the energy, E, of a charged atomic oscillator in the wall of a black body, and the frequency, ν, of its associated electromagnetic wave. Its relevance is now integral to the field of quantum mechanics, describing the relationship between energy and frequency, known as the Planck–Einstein relation:
E = h\nu .
In 1905 the value E, the energy of a charged atomic oscillator, was theoretically associated with the energy of the electromagnetic wave itself, representing the minimum amount of energy required to form an electromagnetic field (a "quantum"). Further investigation of quanta revealed behaviour associated with an independent unit ("particle") as opposed to an electromagnetic wave and was eventually given the term photon. The Planck–Einstein relation now describes the energy of each photon in terms of the photon's frequency. This energy is extremely small in terms of ordinary experience.
The above equation leads to another relationship involving the Planck constant. Given p for the linear momentum of a particle (not only a photon, but other particles as well), the de Broglie wavelength λ of the particle is given by
\lambda = \frac{h}{p} .
In applications where it is natural to use the angular frequency (i.e. where the frequency is expressed in terms of radians per second instead of cycles per second or hertz) it is often useful to absorb a factor of 2π into the Planck constant. The resulting constant is called the reduced Planck constant or Dirac constant. It is equal to the Planck constant divided by 2π, and is denoted ħ (pronounced "h-bar"):
\hbar = \frac{h}{2 \pi} .
The energy of a photon or particle with angular frequency ω, where ω = 2πν, is given by
E = \hbar \omega ,
while its linear momentum relates to
p = \hbar k .
